c Linearly independent as 2 2 7 1 2 3 3 5 9 row reduces to 1 1 1 d Linearly

# C linearly independent as 2 2 7 1 2 3 3 5 9 row

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(c) Linearly independent: as 2 2 7 1 - 2 3 3 - 5 9 row reduces to 1 0 0 0 1 0 0 0 1 . (d) Linearly independent: as A = 1 1 1 . . . 1 0 2 2 . . . 2 0 0 3 · · · 3 . . . . . . . . . . . . . . . 0 0 0 . . . n row reduces to the identity n × n matrix. 11
7. (a) Use Gauss-Jordan elimination to solve the homogeneous system x 1 - x 2 - 4 x 3 = 0 x 1 + x 2 + 4 x 3 = 0 2 x 2 + x 3 = 0 (b) Find by inspection a particular solution of the system x 1 - x 2 - 4 x 3 = - 4 x 1 + x 2 + 4 x 3 = 6 2 x 2 + x 3 = 3 and hence write down the general solution, using (a). Solution (a) We row reduce the corresponding augmented matrix: 1 - 1 - 4 0 1 1 4 0 0 2 1 0 1 - 1 - 4 0 0 2 8 0 0 2 1 0 1 0 0 0 0 1 4 0 0 0 - 7 0 1 0 0 0 0 1 0 0 0 0 1 0 which shows that there is a unique solution x 1 x 2 x 3 = 0 0 0 . (b) By inspection we see that substituting in x 1 = x 2 = x 3 = 1 gives the desired result, hence the general solution is x 1 x 2 x 3 = 0 0 0 + 1 1 1 = 1 1 1 . 8. Let A = 1 5 2 - 1 9 5 1 3 1 and let W be the set of all linear combinations of the columns of A . (a) Show that (4 , 10 , 2) is in W (you should be able to do this by inspection, that is, without any row operations). (b) Solve the homogeneous system with augmented matrix [ A | 0 ]. (c) Given that (1 , 2 , - 1) is a solution of the linear system [ A | b ], what is the vector b ? (d) Write down the general solution of [ A | b ] (with the vector b from part (c)), without directly solving this system. Solution (a) By inspection we see that 4 10 2 is given by the following linear combination of the columns: 0 1 - 1 1 + 0 5 9 3 + 2 2 5 1 . 12
(b) Row reducing gives: 1 5 2 0 - 1 9 5 0 1 3 1 0 1 5 2 0 0 14 7 0 0 - 2 - 1 0 1 5 2 0 0 1 1 2 0 0 0 0 0 1 0 - 1 2 0 0 1 1 2 0 0 0 0 0 Let x 3 = t , then x 1 = 1 2 t and x 2 = - 1 2 t , so ( x 1 , x 2 , x 3 ) = t 1 2 - 1 2 1 , for any t R . (c) The first coordinate of b is 1 × 1 + 5 × 2 + 2 × ( - 1) = 9. The second coordinate of b is - 1 × 1 + 9 × 2 + 5 × ( - 1) = 12. The third coordinate of b is 1 × 1 + 3 × 2 + 1 × ( - 1) = 6. So b = 9 12 6 . (d) The general solution of [ A | b ] is given by adding together the general solution of [ A | 0 ] and a particular solution of [ A | b ]. In this case we have x = 1 2 - 1 + t 1 2 - 1 2 1 , t R . 9. For which value(s) of d are the following sets of vectors linearly dependent? Justify your answers. (a) { (1 , - 1 , 4)) , (3 , - 5 , 5) , ( - 1 , 5 , d ) } (b) { (3 , 7 , - 2) , ( - 6 , d, 4) , (9 , 1 , - 4) } Solution (a) Linearly dependent if and only if d = 10 as 1 3 - 1 - 1 - 5 5 4 5 d row reduces to 1 0 5 0 1 - 2 0 0 - 10 + d . (b) Linearly dependent if and only if d = - 14 as 3 - 6 9 7 d 1 - 2 4 - 4 row reduces to 1 - 2 0 0 d + 14 0 0 0 1 . 10. Determine, with reasons, whether the following statements are (A) always true, (B) always false, or (C) might be true or false. (a) If v 1 , ..., v 5 R 5 and v 1 = v 2 + v 3 , then the set { v 1 , . . . , v 5 } is linearly dependent. (b) If v 1 and v 2 R 2 lie on the same straight line through the origin, then the set { v 1 , v 2 } is linearly dependent.